Partition Equation

Theorem

Let group $G$ act on a finite set $X$.

Let the distinct orbits of $X$ under the action of $G$ be:

$\Orb {x_1}, \Orb {x_2}, \ldots, \Orb {x_s}$

Then:

$\card X = \card {\Orb {x_1} } + \card {\Orb {x_2} } + \cdots + \card {\Orb {x_s} }$

Proof

Follows trivially from the fact that the Group Action Induces Equivalence Relation.

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$\blacksquare$